Question

1. find the value of tan u rounded to the nearest hundredth, if necessary.
2. find the value of sin o rounded to the nearest hundredth, if necessary.
3. find the value of cos z rounded to the nearest hundredth, if necessary.
4. find the exact value of sin k in simplest radical form.
5. find the exact value of tan o in simplest radical form.

Explanation:

Step1: Find side TU (right triangle)

First, calculate the length of side $TU$ using the Pythagorean theorem for $\triangle TUE$:
$$TU = \sqrt{UE^2 - TE^2} = \sqrt{(\sqrt{48})^2 - (\sqrt{39})^2} = \sqrt{48 - 39} = \sqrt{9} = 3$$

Step2: Compute $\tan U$

For angle $U$, $\tan U = \frac{\text{opposite}}{\text{adjacent}} = \frac{TE}{TU}$
$$\tan U = \frac{\sqrt{39}}{3} \approx \frac{6.245}{3} \approx 2.08$$

---

Step1: Identify sides for $\sin O$

In right $\triangle MNO$, for angle $O$: opposite side $MN = \sqrt{78}$, hypotenuse $MO = 10$

Step2: Compute $\sin O$

$$\sin O = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{\sqrt{78}}{10} \approx \frac{8.832}{10} \approx 0.88$$

---

Step1: Find side XY (right triangle)

For $\triangle XYZ$, calculate $XY$ using Pythagorean theorem:
$$XY = \sqrt{XZ^2 - YZ^2} = \sqrt{20^2 - 2^2} = \sqrt{400 - 4} = \sqrt{396} = 6\sqrt{11}$$

Step2: Compute $\cos Z$

For angle $Z$, $\cos Z = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{YZ}{XZ}$
$$\cos Z = \frac{2}{20} = 0.10$$

---

Step1: Identify sides for $\sin K$

In right $\triangle KLM$, for angle $K$: opposite side $LM = 4$, hypotenuse $KM = 10$

Step2: Simplify $\sin K$

$$\sin K = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{4}{10} = \frac{2}{5}$$

---

Step1: Identify sides for $\tan O$

In right $\triangle MNO$, for angle $O$: opposite side $MN = \sqrt{8}$, adjacent side $ON = 7$

Step2: Simplify $\tan O$

$$\tan O = \frac{\text{opposite}}{\text{adjacent}} = \frac{\sqrt{8}}{7} = \frac{2\sqrt{2}}{7}$$

Answer:

1. $\approx 2.08$
2. $\approx 0.88$
3. $0.10$
4. $\frac{2}{5}$
5. $\frac{2\sqrt{2}}{7}$